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We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matricesand descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359]that, for any n, k, m and p, the number of nxn alternating sign matrices (ASMs)for which the 1 of the first row is in column k+1 and there are exactly m -1'sand m+p inversions is equal to the number of descending plane partitions (DPPs)for which each part is at most n and there are exactly k parts equal to n, mspecial parts and p nonspecial parts. The proof involves expressing theassociated generating functions for ASMs and DPPs with fixed n as determinantsof nxn matrices, and using elementary transformations to show that thesedeterminants are equal. The determinants themselves are obtained by standardmethods: for ASMs this involves using the Izergin-Korepin formula for thepartition function of the six-vertex model with domain-wall boundaryconditions, together with a bijection between ASMs and configurations of thismodel, and for DPPs it involves using the Lindstrom-Gessel-Viennot theorem,together with a bijection between DPPs and certain sets of nonintersectinglattice paths.

On the weighted enumeration of alternating sign matrices and descending plane partitions